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Chapter 5: Problem 28

A racetrack has the shape of an inverted cone, as the drawing shows. On this surface the cars race in circles that are parallel to the ground. For a speed of 34.0 m/s, at what value of the distance \(d\) should a driver locate his car if he wishes to stay on a circular path without depending on friction?

Short Answer

Expert verified
Use \\( d = \frac{v^2}{g \tan(\theta)} \\) with given speed and cone angle to find \\( d \\).

Step by step solution

01

Understand the Problem

The problem describes a car on an inverted cone-shaped racetrack. The car travels in a circular path parallel to the ground, and friction is not considered. We need to find the distance from the center of the cone's base to where the car should be located while traveling at 34.0 m/s.
02

Identify the Relevant Forces

Since the car is moving in a circle at constant speed, there must be a centripetal force acting towards the center of the circle. The forces involved include the gravitational force acting downwards and the normal force acting perpendicular to the cone's surface. The normal force provides the necessary centripetal force for circular motion.
03

Set Up the Equations of Motion

Assume the car's mass is \( m \), the gravitational acceleration is \( g \), and the angle between the cone's surface and the horizontal is \( \theta \). The component of gravitational force perpendicular to the surface is \( mg \cos(\theta) \). By balancing the forces, the normal force \( N \) provides the centripetal force: \[ N \sin(\theta) = \frac{mv^2}{d} \] where \( d \) is the radius of the path.
04

Use Newton's Second Law Vertically

In the vertical direction, there is no movement, so forces balance: \[ N \cos(\theta) = mg \]. This equation allows us to express \( N \) in terms of \( m \) and \( g \): \[ N = \frac{mg}{\cos(\theta)} \].
05

Solve for Distance \( d \)

Substitute the expression for \( N \) derived from the vertical forces into the centripetal force equation: \[ \frac{mg \sin(\theta)}{\cos(\theta)} = \frac{mv^2}{d} \].Simplify and solve for \( d \): \[ d = \frac{v^2}{g \tan(\theta)} \].
06

Calculating the Distance \( d \)

With given speed \( v = 34.0 \text{ m/s} \) and assuming \( g = 9.8 \text{ m/s}^2 \), find the angle \( \theta \) from the problem setup or given data. If \( \theta \) or \( \tan(\theta) \) is known, substitute to find \( d \): \[ d = \frac{(34.0)^2}{9.8 \tan(\theta)} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centripetal Force
In circular motion, a centripetal force is crucial. It is the force that keeps an object moving in a curve by acting towards the center of the circle. Without this force, an object would travel in a straight line due to inertia.
In the context of our problem, the car on the track needs a centripetal force to maintain its circular path. This force doesn't come from friction, as we assumed it's negligible. Instead, the track's surface provides it through the normal force, which we'll learn about shortly.
The formula for centripetal force is \[ F_c = \frac{mv^2}{r} \] where:
  • \( F_c \) = centripetal force
  • \( m \) = mass of the object
  • \( v \) = velocity of the object
  • \( r \) = radius of the circular path
This force ensures the car doesn't drift off the track.
Normal Force
The normal force is one that acts perpendicular to a surface. It is what prevents objects from passing through solid surfaces.
In our scenario, the normal force acts perpendicular to the conical surface of the racetrack. However, it not only keeps the car from sinking into the track but also provides the centripetal force needed for circular motion.
This is because the component of the normal force along the surface provides the necessary inward pull:\[ N \sin(\theta) = \frac{mv^2}{d} \]where \( N \) is the normal force and \( d \) is the required radius to maintain the car's path. It's essential for keeping the car stable as it races around the track.
Newton's Second Law
Newton's Second Law is foundational in physics, defining how forces affect motion. It states that the force acting on an object is equal to the mass of that object times its acceleration: \[ F = ma \].
This principle allows us to comprehend how multiple forces interact and balance.
In the context of the racing car, we consider forces in the
  • Horizontal direction: Where the centripetal force must equal the horizontal component of the normal force.
  • Vertical direction: Where forces must balance (i.e., no vertical movement), leading to \( N \cos(\theta) = mg \) to find the magnitude of the normal force.
Understanding this law helps us solve for the distance \( d \) from the center of the cone to the vehicle.
Gravitational Force
Gravitational force is the force of attraction between two masses. On Earth, it gives weight to objects and pulls them towards the center of the planet.
For our racecar, gravitational force acts downward, represented by \[ F_g = mg \], where:
  • \( F_g \) = gravitational force
  • \( m \) = mass of the car
  • \( g \) = acceleration due to gravity (9.8 m/s2)
This force interacts with other forces at play. We must consider its component that acts perpendicular to the track's surface, influencing the normal force and playing a key role in maintaining the car's circular path through its action perpendicular to the conical surface.

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Most popular questions from this chapter

A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to twice her weight as she goes through the dip. If r 20.0 m, how fast is the roller coaster traveling at the bottom of the dip?

Multiple-Concept Example 7 deals with the concepts that are important in this problem. A penny is placed at the outer edge of a disk (radius 0.150 m) that rotates about an axis perpendicular to the plane of the disk at its center. The period of the rotation is 1.80 s. Find the minimum coefficient of friction necessary to allow the penny to rotate along with the disk.

Pilots of high-performance fighter planes can be subjected to large centripetal accelerations during high-speed turns. Because of these accelerations, the pilots are subjected to forces that can be much greater than their body weight, leading to an accumulation of blood in the abdomen and legs. As a result, the brain becomes starved for blood, and the pilot can lose consciousness (鈥渂lack out鈥�). The pilots wear 鈥渁nti-G suits鈥� to help keep the blood from draining out of the brain. To appreciate the forces that a fighter pilot must endure, consider the magnitude \(F_{N}\) of the normal force that the pilot's seat exerts on him at the bottom of a dive. The magnitude of the pilot's weight is \(W .\) The plane is traveling at 230 \(\mathrm{m} / \mathrm{s}\) on a vertical circle of radius 690 \(\mathrm{m}\) . Determine the ratio \(F_{N} / W\) . For comparison, note that blackout can occur for values of \(F_{\mathrm{N}} / W\) as small as 2 if the pilot is not wearing an anti-G suit.

At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of 3.2 m/s, and an 83-kg person feels a 560-N force pressing against his back. What is the radius of the chamber?

A satellite moves on a circular earth orbit that has a radius of \(6.7 \times 10^{6} \mathrm{~m} .\) A model airplane is flying on a \(15-\mathrm{m}\) guideline in a horizontal circle. The guideline is parallel to the ground. Find the speed of the plane such that the plane and the satellite have the same centripetal acceleration.
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